Our interest in the problem of permeability measurement arises out of the closed-cell foam insulation industry. Closed-cell foam products have been used to great advantage in a wide variety of industrial applications due to their extremely low thermal conductivity, but they have always been subject to a little understood and poorly quantified aging phenomenon in which a foam's thermal properties degrade over time. Our objective has been to quantify the speed of the aging process and thereby assist industry in developing foams which are more resistant to this process.
The insulating value of a closed-cell foam depends largely on the thermal conductivity of the gas mixture which fills the cells of the foam. When the foam is first produced, the cells are filled with the blowing agent, usually a fluorocarbon gas, which has a low thermal conductivity (typically about one-third of that for stagnant air). As the foam is exposed to the environment over time, the component gases of air permeate into the cells of the foam and gradually dilute the fluorocarbon gas, producing a mixture which has a higher thermal conductivity than the fluorocarbon alone. Eventually the fluorocarbon gas permeates out of the foam, causing a further increase in the thermal conductivity of the remaining gas mixture. As this aging process occurs, the overall thermal conductivity of the foam rises and the insulating value of the foam drops proportionately. The time scales of this aging process can be quantified if the rates of permeation, in the form of the respective "permeability coefficient", of the air components and the fluorocarbon blowing agents are known. The two central components of permeability are the diffusion coefficient, which quantifies transient transport processes, and the solubility coefficient, which characterizes steady-state storage capacity.
The main difficulty in determining these permeability coefficients is the amount of time required to obtain accurate measurements. In systems with low diffusion rates, the measurement of these coefficients is extremely time-consuming. In systems with low solubilities, the measuring apparatus must be very sensitive to produce meaningful results. Closed-cell foam insulation is an example of a system with both low diffusion rates and low solubilities and it therefore presents one of the most challenging applications for these types of measurements. Typical times required to measure the permeability of one gas species in one foam sample have been on the order of weeks to months using currently available techniques. Polymer chemists developing new aging-resistant foams need this data as quickly as possible to make the most useful progress.
One drawback of these foam products is their use of chlorofluorocarbons, or CFCs, which have recently been linked to the degradation of the Earth's ozone layer. The foam manufacturing process relies on the unique thermodynamic properties of the CFC to act as a "blowing agent", which expands and inflates the polymeric mixture as the mixture changes phase from liquid to solid, thereby forming the cellular foam structure. This process produces a foam whose cells are filled with CFC vapor. During the service life of the foam or when the foam is destroyed, this vapor may be released into the atmosphere and subsequently contribute to the ozone degradation. Fortunately this problem has received attention on an international scale which has led to a scheduled phasing out of the use of the CFC products which are known to be harmful. Researchers in the industry have responded with efforts to find replacements for the banned CFCs. To aid this research for a replacement, industry needs a tool for more rapidly and accurately assessing the characteristics of new blowing agents and the related foam products.
1. Industry Accelerated Aging Test
In order to quantify the relative aging rates of closed-cell foams, the industry has attempted to use the Accelerated Aging Test, so called because it takes advantage of the increase in permeation rates with temperature to accelerate the aging process. In the test, a fresh foam sample is maintained in a 40.degree. C. environment for 90 to 180 days. The overall thermal conductivity of the sample is measured before and after this exposure and the difference is taken as an indication of the foam's resistance to aging.
This test is useful to industry because it distinguishes foam aging characteristics on a relative scale, but from a broader point of view the test has several major drawbacks. The central problem is that the effect of temperature on aging rates is not known exactly, so it is difficult to relate the aging that occurs during the accelerated test to that which might occur in some specified operating conditions other than a constant 40.degree. C. For example, Ostrogorsky, A. G., "Aging of Polyurethane Foams" Ph.D. Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, 1985, recently found that the increase in the permeation rate of the blowing agent for a given increase in temperature is larger than that of the air components. The test also does not give any insight into the physics of the aging process which might enable the development of superior foams. Furthermore, although it is accelerated relative to the normal aging time scale, it is still too slow to provide timely feedback on foam quality in attempts to improve foam aging characteristics by empirical means.
2. General Model of Foam Aging Process
The first step in understanding the foam aging process is the development of a model which quantifies the relationship between gas permeation and changes in thermal conductivity. Cuddihy, E. F., and Moacanin, J., "Diffusion of Gases in Polymeric Foams", JournaI of Cellular Plastics, vol. 3, Feb. 1963, pp. 73-80, first demonstrated that closed-cell foam can be treated as a homogeneous medium which obeys Fick's first law of diffusion (see Brehm Thesis, Section 2.1.3). If the diffusion coefficient is assumed to be independent of concentration, then the time-dependent gas concentration in the foam may be modelled by Fick's second law with uniform initial conditions and constant boundary conditions. The solution of this model was first given by Newman, A. B., "The Drying of Porous Solids: Diffusion Calculations", Transactions of the American Institute of Chemical Engineers, 27, 1931, p.310, for a slab geometry and has been presented by Carslaw, H. S., and Jaeger, J. S., The Conduction of Heat in Solids, Oxford University Press, Oxford, England, 1959, and by Arpaci, V. S., Conduction Heat Transfer, Addison-Wesley Publishing Co., Reading, MA, 1966 for a variety of other geometries and initial conditions. If the diffusion coefficient is known for each gas, these solutions may be used to predict the variation in a foam's cell gas composition with time.
A review of models for the thermal conductivity of gas compositions is given by Tsederberg, N. V., Thermal Conductivity of Gases and Liquids, The MIT Press, Cambridge, MA, 1965. Schuetz, M. A., "Heat Transfer in Foam Insulation", M. S. Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, 1982, used an equation with empirical corrections proposed by Lindsey and Bromley, presented by Tsederberg, to calculate the thermal conductivity of the cell gas mixture. He proposed calculating the thermal conductivity of the gas composition at several positions in the foam to find an effective thermal conductivity for the foam gas mixture.
Ostrosgorsky, supra, presented a numerical simulation of Fick's law applied to a foam panel in which the initial gas concentrations may be prescribed and a temperature profile may be imposed. He followed Schuetz in calculation of the thermal conductivity of the cell gas mixture and assumed an Arrhenius-type variation of gas permeability with temperature. If the constants from this Arrhenius equation are given for each gas, the simulation predicts the increase in the overall thermal conductivity of a foam panel as a function of time.
It is important to realize that each of the models requires the diffusion and solubility coefficients of the air components and the blowing agent in the foam to predict the rate of thermal aging.
3. Geometric Models and Measurement Techniques
Since the polymer cell walls are the controlling factor in the gas permeation process, much effort has been focussed on measuring the geometric characteristics of the cell structure and modelling the effects which this structure has on the overall permeability of the foam. The first model of this type was developed by Cuddihy and Moacanin, supra, and is based on one-dimensional flow through a cubical cell structure with uniform wall thickness. Valenzuela and Glicksman, "Thermal Resistance and Aging of Rigid Urethane Foam Insulation, Proceedings of DOE-ONRL Workshop on Mathematical Modelling of Roofs, Atlanta, GA, CONF-811179, Nov. 3-4, 1981, p. 261, showed that slight changes in this geometry, e.g. staggering the cells, may change the results by as much as 100%.
The first quantitative characterization of foam cell structure was done by Reitz, D. W., "A Basic Study of Gas Diffusion in Foam Insulation", M. S. Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, 1983, using an innovative sectioning technique. Based on his measurements and some correlation with permeability measurements, he concluded that cell wall thickness, polymer permeability, and foam density are the most important parameters for predicting foam aging, while the effects of assumed cell structure are of secondary importance. He also demonstrated that only 10-20% of the solid polymer in a typical polyurethane foam is in the cell walls, with the remaining 80-90% in the struts which form at the intersections of cell walls. Ostrogorsky, supra, refined Reitz's model by adding a geometrical enhancement parameter and verified the model using a two-dimensional electrical analogy. His results correlate well with his measurements of permeability in foams and in polymer films which were assumed to be representative of cell walls. Fox, T. J., "Aging of Closed-Cell Phenolic Foam", M. S. Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, 1986, applied a similar model to the study of phenolic foams and used Scanning Electron Microscope photography to improve the geometric measurement technique.
Two particularly valuable results have come from this study of the microscopic structure of the foam. The first is the ability to make an estimate of foam permeability based on measurements of cell wall permeability and the relatively straightforward measurements of the foam geometry. This is especially useful when comparing the performance of foams which have similar polymer composition but may, for example, vary in cell size or cell wall thickness. The second valuable result of the geometric models is the ability to translate from the permeability of the foam to the permeability of the cell wall, and vice versa. This gives the experimenter the freedom to measure either quantity and readily obtain insight into the other as well. As permeability measurements of either the foam or the polymer are very difficult and time-consuming, any added flexibility such as this can be very advantageous.
The gas permeation models and geometric models all rely on the ability to measure either the permeability of the solid polymer or the effective permeability of the foam. Because of extremely low permeation rates, especially for the CFC blowing agents, these measurements have proven rather difficult and extremely time-consuming. The large variation in the published data for foam effective diffusion coefficients, shown in Table 1, implies the need for more accurate measurement techniques, while the scarcity of this type of data indicates the need for faster test methods.
TABLE 1 ______________________________________ Summary of Published Data for Effective Diffusion Coefficients of O.sub.2, N.sub.2, and R11 in Urethane Foams at 25 C. FOAM DIFFUSION COEFFICIENT DENSITY [10.sup.-8 cm.sup.2 /sec] [kg/m.sup.3 ] O.sub.2 N.sub.2 R11 SOURCE* ______________________________________ 33.5 200. 100. 3.0 Ball 35.9 4.0-7.0 5.5-16. .006-.25 Brandreth 33.5 6.0 2.0-3.0 .025-.046 Brandreth 35.2 11.2 6.3 .225 Norton 35.2 -- -- .042 Norton 24.5 196. -- -- Lee 25.6 148. -- -- Lee 20.7 76.8 -- -- Reitz 28.3 119. -- -- Reitz 30.4 193. -- -- Reitz 25.2 46.8* 7.6 .23-.57 Ostrogorsky ______________________________________ *References: Ball, G. W., Hard, R., Walker, M. G., "The Thermal Conductivity of Rigid Urethane Foams", Journal of Cellular Plastics, 1970 Brandreth, D. A., Ingersole, H. G., "Accelerated Aging of Rigid Polyurethane Foam", Unpublished Report, E. I. Dupont de Nemours and Co., Wilmington, Delaware; Norton, F. J., "Diffusion of Chlorofluorocarbon Gases in Polymer Films and Foams", Journal of Cellular Plastics, 1982, p. 300; Lee, W. M., Brown, C. N., "Gas Permeability Determination in Urethan Foams", Journal of Thermal Insulation, vol. 6, 1983; Reitz, supra; Ostrogorsky, supra.
4. Current Permeability Measurement Techniques: Transmission vs. Sorption/Desorption
Currently the most commonly used techniques of computing gas permeability through closed-cell foam insulation can be divided into two categories: transmission methods, and sorption/desorption methods. The latter method is used in the present invention.
In the transmission method, the material to be tested is placed between two isolated chambers which contain different pressures of the test gas. When the gas flow rate from the high pressure side to the low pressure side has reached a steady state, this flow rate is measured and the value is used to calculate the desired permeability coefficient. The gas flow is generally measured by recording either a constant-pressure volume change, or a constant-volume pressure change on the low pressure side of the material. Such a transmission system is basically described in S. K. Roy, "Permeability Cell", U.S. Pat. No. 3,548,634, Dec. 22, 1970, as applied to ceramics for the purpose of obtaining information on its internal pore characteristics. Other detection techniques have been used in transmission methods, including laser spectroscopy, gas chromatography and flame ionization, but most of these are expensive, cumbersome, and limited in application.
With a sorption/desorption method as practiced in the prior art, the material to be tested is placed in a chamber in which it is surrounded by the test gas. Before the test begins, the concentration of the gas inside the material is allowed to come to equilibrium with the pressure of the gas in the chamber. To begin the test, the pressure of the surrounding gas is suddenly changed (increased for sorption, decreased for desorption), and the rate at which the material takes up or gives off the test gas is recorded. From the measured sorption/desorption rate, the diffusion coefficient is determined, and from the equilibrium gas uptake the solubility coefficient is found. In most cases the gas uptake is recorded by "gravimetric" means, in which the pressure of the test gas around the material is held constant while the increase in weight of the sample is recorded.
The general concept of a constant-volume sorption method of measuring diffusion coefficients appears to have been first proposed by Carman and Haul, in Proceedings of the Royal Society of London, Section A, volume 222 , pp. 109-118 (1954). Their paper suggests the general method and discusses some of the mathematical analysis, but it gives no structure whatever for implementing this method or reducing it to practice. That paper presents some data obtained solely in using the method to measure gas adsorption in solid silica material with moderate permeation rates.